Pexiderized homogeneity almost everywhere

In the paper we examine Pexiderized phi-homogeneity equation almost everywhere. Assume that G and H are groups with zero, (X, G) and (Y, H) are a G- and an H-space, respectively. We prove, under some assumption on (Y, H), that if functions phi: G -> H and F(1), F(2): X -> Y satisfy Pexiderized phi-homogeneity equation F(1) (alpha x) = phi(alpha) F(2) (x) almost everywhere in G x X then either phi = 0 almost everywhere in G or F(2) = 0 almost everywhere in X or there exists a homomorphism phi: G -> H such that phi = a phi almost everywhere in G and there exists a function F: X -> Y such that (F) over bar(alpha x) = phi(alpha)(F) over bar (x) for alpha is an element of G\{0}, x is an element of X, and F(1) = a (F) over bar almost everywhere in X, F(2) = (F) over bar almost everywhere in X, where a is an element of H* is a constant. From this result we derive solution of the classical Pexider equation almost everywhere. (c) 2006 Elsevier Inc. All rights reserved.